An excursion into the string spectrum

TL;DR

This paper introduces a covariant method using Howe duality to systematically explore the spectrum of physical states in bosonic string theory through entire trajectories, revealing new structural insights and enabling amplitude computations.

Contribution

It presents a novel covariant framework based on Howe duality to analyze string states as trajectories, extending the understanding of string spectra beyond individual states.

Findings

Identifies infinite trajectories of string states as lowest-weight representations.

Develops trajectory-shifting operators to probe deeper states.

Calculates sample tree-level amplitudes for subleading trajectories.

Abstract

We propose a covariant technique to excavate physical bosonic string states by entire trajectories rather than individually. The approach is based on Howe duality: the string's spacetime Lorentz algebra commutes with a certain inductive limit of $sp(\bullet)$, with the Virasoro constraints forming a subalgebra of the Howe dual algebra $sp(\bullet)$. There are then infinitely many simple trajectories of states, which are lowest-weight representations of $sp(\bullet)$ and hence of the Virasoro algebra. Deeper trajectories are recurrences of the simple ones and can be probed by suitable trajectory-shifting operators built out of the Howe dual algebra generators. We illustrate the formalism with a number of subleading trajectories and compute a sample of tree-level amplitudes.